Applications of the Backus-Gilbert method to linear and some non linear equations

TL;DR

This paper explores a functional analytical adaptation of the Backus-Gilbert method to reconstruct solutions of linear and mildly non-linear equations, providing error estimates and applications to specific problem classes.

Contribution

It introduces a novel functional analytical approach to the Backus-Gilbert method for linear and non-linear systems, including error analysis and applications to moment and elliptic Cauchy problems.

Findings

Error estimates for the reconstruction method

Application to nonlinear moment problems

Analysis of elliptic Cauchy problems using Green-formula

Abstract

We investigate the use of a functional analytical version of the Backus-Gilbert Method as a reconstruction strategy to get specific information about the solution of linear and slightly non-linear systems with Frech\'et derivable operators. Some a priori error estimates are shown and tested for two classes of problems: a nonlinear moment problem and a linear elliptic Cauchy problem. For this second class of problems a special version of the Green-formula is developed in order to analyze the involved adjoint equations.